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A271025 Array read by antidiagonals: T(i,j) is the i-th binomial transform of the Catalan sequence (A000108) evaluated at j. 2
1, 1, 1, 2, 2, 1, 5, 5, 3, 1, 14, 15, 10, 4, 1, 42, 51, 37, 17, 5, 1, 132, 188, 150, 77, 26, 6, 1, 429, 731, 654, 371, 141, 37, 7, 1, 1430, 2950, 3012, 1890, 798, 235, 50, 8, 1, 4862, 12235, 14445, 10095, 4706, 1539, 365, 65, 9, 1, 16796, 51822, 71398, 56040 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Interestingly, the determinant of the n X n array of entries of the form T(i,j) is equal to the (n-1)-th superfactorial number (see A000178).

As indicated in A104455, the k-th binomial transform of A000108 will have g.f. (1-sqrt((1-(k+4)x)/(1-kx)))/(2x), e.g.f. exp((k+2)x)(BesselI(0,2x)-BesselI(1,2x)) and a(n)=sum{i=0..n, C(n,i) C(i) k^(n-i)}.

The columns of this array are polynomial integer sequences. The successive polynomials corresponding to the columns of this array are: p0(n)=1, p1(n)=n+1, p2(n)=n^2+2n+2, p3(n)=n^3+3*n^2+6*n+5, p4(n) = n^4+4*n^3+12*n^2+20*n+14, and so forth. The coefficients of these successive polynomials form the following number triangle, which is given by the sequence A098474:

1

1, 1

1, 2, 2

1, 3, 6, 5

1, 4, 12, 20, 14

...

LINKS

Table of n, a(n) for n=0..58.

FORMULA

T(0,j) = A000108(j).

For i>=1, T(i,j) = sum(k=0..j,binomial(j,k)*T(i-1,k)).

T(i,j) = sum(k=0..j, binomial(j,k)*A000108(k)*i^(j-k)).

EXAMPLE

The array given by integers of the form A(i,j) is illustrated below:

[1 1 2  5   14   42    ...]

[1 2 5  15  51   188   ...]

[1 3 10 37  150  654   ...]

[1 4 17 77  371  1890  ...]

[1 5 26 141 798  4706  ...]

[1 6 37 235 1539 10392 ...]

[. . .  .   .    .     .  ]

[. . .  .   .    .      . ]

[. . .  .   .    .       .]

MATHEMATICA

A000108[n_]:= Binomial[2*n, n]/(n+1) ;

T[i_, j_]: Sum[Binomial(j, k)*A000108(k)*i^(j-k), {k, 0, j}] ;

PROG

(Sage) def A000108(n): return binomial(2*n, n)/(n+1) ;

def T(i, j): return sum(binomial(j, k)*A000108(k)*i^(j-k) for k in range(j+1))

CROSSREFS

Cf. A098474, A000178. Rows 0-5: A000108, A007317, A064613, A104455, A104498, A154623. Columns 0-3: A000012, A000027, A002522, A005491.

Sequence in context: A033184 A171567 A110488 * A134379 A108087 A123158

Adjacent sequences:  A271022 A271023 A271024 * A271026 A271027 A271028

KEYWORD

nonn,tabl,easy

AUTHOR

John M. Campbell, Mar 28 2016

STATUS

approved

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Last modified October 17 05:23 EDT 2018. Contains 316275 sequences. (Running on oeis4.)